Chapter 03-Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 3 Analytic Functions Page |1

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
Chapter 3
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


Note: Solutions not appearing in this complete solutions manual can be found in the student
study guide. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
Exercises 3.1
z−z
2.© Jones
lim& Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
z →1+ i z + z
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
z − z x + iy + x + iy y
=f ( z) = = i
z + z x + iy + x − iy x

© Jones & BartlettAccording
Learning,toLLC theorem 3.1.1: © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
z−z
lim =L= u0 + iv0 , where u0 = lim θ = 0
z →1+ i z + z ( x , y ) → (1,1)

y
and
= v
© Jones & Bartlett Learning, LLC x 1
0 lim
= © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION ( x , y ) → (1,1)
NOT FOR SALE OR DISTRIB

z−z
Thus lim =i
z →1+ i z + z

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT
3. FOR
In order to use Theorem 3.1.1 we need to find the realNOT
SALE OR DISTRIBUTION FOR SALE
and imaginary ORofDISTRIBUTION
parts f(z) = |z|2 –
iz . Setting z = x + iy we obtain
2 2
z −LLC
© Jones & Bartlett Learning, iz = x + iy − i ( x + iy ) © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION = x 2 + y 2 − ix − y NOT FOR SALE OR DISTRIBUTION
= ( x 2 + y 2 − y ) + ix.

© Jones
Identifying x0 = 1, &
y0 Bartlett
= –1, u(x,Learning,
y) = x2 + y2LLC © Jones & Bartlett Learning,
– y, and v(x, y) = x, we have
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
2 2
lim ( x + y − y ) 3 =
= and lim x 1.
( x , y ) → (1, −1) ( x , y ) → (1, −1)



© Jones & Bartlett Learning, LLC 2 © Jones & Bartlett Learning, LLC
Therefore,
NOT FOR SALEby Theorem
OR 3.1.1, lim ( z − iz ) =3 − i. NOT FOR SALE OR DISTRIBUTION
DISTRIBUTION z →1−i




© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 3 Analytic Functions Page |2

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
2
NOT FOR SALE Im( z ) NOT FOR SALE OR DISTRIBUTION
4. ORlim DISTRIBUTION
z →3i z + Re( z )


Im( z 2 ) 2 xy 2 xy
f ( z)
= = =
z©+Jones
Re( z ) &xBartlett
+ iy + x Learning,
2 x + iy LLC © Jones & Bartlett Learning,
NOT FOR
2 xy (2 x − iy ) SALE 2OR DISTRIBUTION
4x y 2 xy 2 NOT FOR SALE OR DISTRIB
= = i
4 x2 + y2 4 x2 + y2 4 x2 + y2
4 x2 y
= u ( x, y ) = u0 lim = u ( x, y ) 0
© Jones & Bartlett 4 x 2 + Learning,
y2 LLC
( x , y )→(0,3) © Jones & Bartlett Learning, LLC
NOT FOR SALE2OR 2 DISTRIBUTION
x y NOT FOR SALE OR DISTRIBUTION
= v ( x, y ) = v0 lim
= v ( x , y ) 0
4 x2 + y2 ( x , y )→(0,3)

Im( z′)
⇒ L = lim = 0 + i0 = 0
z →3i
© Jones & Bartlett Learning, z LLC
+ Re( z ) © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

6. lim ze z
z →i

z x +iy
f ( z ) ==
ze xe xLearning,
( x + iy&)eBartlett
© Jones = cos y + ixe xLLC
sin y + iye x cos y − ye x sin©y Jones & Bartlett Learning,
x
f ( z )= eNOT
( x cos
FORy − SALE + ie xDISTRIBUTION
y sin y ) OR ( x sin y + y cos y ) NOT FOR SALE OR DISTRIB
  
u( x, y ) v ( x, y )
u0 = e0 (0 ∗ cos1 − 1sin1) =
lim u( x, y ) = −e0 sin1
( x , y )→(0,1)
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
=v0 lim v ( x,= − e0 cos1
y ) e0 (0 ∗ sin1 + 1cos1) =
NOT FOR SALE
( x , y )→(0,1) OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
L =+ − e0 sin1 + ie0 cos1
u0 iv0 =
lim ze z =
− sin1 + i cos1  −0.8415 + 0.5403i
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE
7. OR InDISTRIBUTION
order to use Theorem 3.1.1 we needNOT FOR
to find the SALE
real andOR DISTRIBUTION
imaginary parts of f ( z ) =

ez − ez
. Setting z = x + iy we obtain
Im( z )
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
eNOT
z
− e z FORe x +SALE
iy
− e x −iyOR DISTRIBUTION NOT FOR SALE OR DISTRIB
=
Im( z ) y
e x cos y + ie x sin y − e x cos y + ie x sin y
=
© Jones & Bartlett Learning, LLC y © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
x  sin y 
= 2e   i.
 y 

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 3 Analytic Functions Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR  sin y 
NOT FOR SALE OR DISTRIBUTION
Identifying x0 = 0, y0 = 0, u(x, y) = 0, and v(x, y) =SALE
2ex  OR DISTRIBUTION
, we have
 y 


=© lim
Jones
( x , y ) → (0,0)
0 &0 Bartlett = lim 2eLLC
and Learning,
( x , y ) → (0,0)
x  sin y 

 y 
 2=
x →0

 y →0
(
sin y 
lim e x  lim© Jones
y 
 2. )
& Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
sin y
In the limit of v( x, y ) we made use of the fundamental trigonometric limit lim = 1.
y →0 y
© Jones & Bartlett Learning, LLC 2 © Jones & Bartlett Learning, LLC
Therefore, by Theorem 3.1.1, lim ( z − iz ) =
2i.
NOT FOR SALE OR DISTRIBUTION z →1−i NOT FOR SALE OR DISTRIBUTION



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION
 y NOT FOR SALE OR DISTRIBUTION
8. lim  log e x 2 + y 2 + i arctan 
z →C
 x
y
z ) log e x 2 + y 2 + ia arctan
f (=
©
 Jones
 & Bartlett
  x
Learning, LLC © Jones & Bartlett Learning,
NOTu ( x ,FOR
y) SALE v ( xOR
, y ) DISTRIBUTION NOT FOR SALE OR DISTRIB
u0 lim
= u ( x, =
y ) log e 0 +=
1 log=
e1 0
( x , y ) = (0,1)

 π / 2 if x → 0
= v0 &lim
© Jones = v ( xLearning,
Bartlett , y)  LLC © Jones & Bartlett Learning, LLC
( x , y ) = (0,1)
NOT FOR SALE OR DISTRIBUTION  −π / 2 if x → 0 NOT FOR SALE OR DISTRIBUTION
⇒ L does not exist



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
10. lim( z 5 − z 2 + z=
) lim z 5 − lim z 2 − lim z
z →C z →i z →i z →i


( ) ( )
5
= lim z − lim z + lim z
z →i z →i z →i
© Jones 5& Bartlett
2
Learning, LLC © Jones & Bartlett Learning,
= i −SALE
NOT FOR i + i OR DISTRIBUTION NOT FOR SALE OR DISTRIB
= i +1+ i
= 2i + 1

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 3 Analytic Functions Page |4

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
11. OR
NOT FOR SALE ByDISTRIBUTION
(15), we have NOT FOR SALE OR DISTRIBUTION
lim z = eiπ /4 .
z → eiπ /4


Using this limit, Theorem
© Jones 3.1.2(ii),
& Bartlett and Theorem
Learning, LLC 3.1.2(iv), we obtain© Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
 1 1
limiπ /4  z +  = eiπ /4 + iπ /4
z →e  z e
= eiπ /4 + e − iπ /4
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR = cos(π / 4) + i sin(π / 4) + cos(−πNOT
DISTRIBUTION / 4) +FOR −π / 4) OR DISTRIBUTION
i sin(SALE
= 2.


( lim z ) + 1©NOT
2
© Jones & Bartlett Learning, LLC( z 2 + 1) Jones & Bartlett Learning, LLC
z 2 + 1 zlim z →1+ i
12. OR
NOT FOR SALE =limDISTRIBUTION =→1+ i FOR SALE OR DISTRIBUTION
( lim z ) −1
z →1+ i z 2 − 1 2
lim ( z 2 − 1)
z →1+ i
z →1+ i
2
(1 + i ) + 1 2i + 1 3 − 4i
=
2
=
(1 + i )&
© Jones 1 2i − 1Learning,
−Bartlett 5 LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB

14. lim
z 2 − (2 + i ) 2
= lim
( z − (2 + i) )( z + (2 + i) )
z → 2 + i z − (2 + i ) z →2+i ( z − (2 + i) )
© Jones & Bartlett = Learning,
lim ( z +LLC
2 + i) © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION z →2+i NOT FOR SALE OR DISTRIBUTION
= lim z + 2 + i
z →2+i

= 2 + i + 2 + i = 4 + 2i

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
15. By (15), (16),
NOT FOR SALE OR DISTRIBUTION and Theorem 3.1.2(ii), we
NOThave
FOR SALE OR DISTRIBUTION

lim( z − z0 ) =
0.
z → z0



Thus, we©cannot
Jonesapply
& Bartlett
Theorem Learning, LLC simplifying the rational
3.1.2(iv) without © Jones & Bartlett
function in the Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
limit. Notice that:

(az + b) − (az0 + b) az + b − az0 − b
=
© Jones & BartlettzLearning,
− z0 LLC z − z0 © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION a ( z − z0 ) NOT FOR SALE OR DISTRIBUTION
= .
z − z0


© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.